Question: Find all numbers in the range of
\[f(x) = \arctan x + \arctan \frac{1 - x}{1 + x},\]expressed in radians.  Enter all the numbers, separated by commas.
Explanation: From the addition formula for tangent,
\[\tan f(x) = \tan \left( \arctan x + \arctan \frac{1 - x}{1 + x} \right) = \frac{x + \frac{1 - x}{1 + x}}{1 - x \cdot \frac{1 - x}{1 + x}} = 1.\]If $x < -1,$ then $-\frac{\pi}{2} < \arctan x < -\frac{\pi}{4}.$  Also,
\[1 + \frac{1 - x}{1 + x} = \frac{2}{1 + x} < 0,\]so $\frac{1 - x}{1 + x} < -1,$ which means $-\frac{\pi}{2} < \arctan \frac{1 - x}{1 + x} < -\frac{\pi}{4}.$  Therefore, $-\pi < f(x) < -\frac{\pi}{2}.$  Since $\tan f(x) = 1,$ $f(x) = -\frac{3 \pi}{4}.$

If $x > -1,$ then $-\frac{\pi}{4} < \arctan x < \frac{\pi}{2}.$  Also,
\[1 + \frac{1 - x}{1 + x} = \frac{2}{1 + x} > 0,\]so $\frac{1 - x}{1 + x} > -1,$ which means $-\frac{\pi}{4} < \arctan \frac{1 - x}{1 + x} < \frac{\pi}{2}.$  Therefore, $-\frac{\pi}{2} < f(x) < \pi.$  Since $\tan f(x) = 1,$ $f(x) = \frac{\pi}{4}.$

Therefore, the range of $f(x)$ consists of the numbers $\boxed{-\frac{3 \pi}{4}, \frac{\pi}{4}}.$